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We study photoproduction with various hadronic rescattering contributions included, in addition to the Pomeron and pseudoscalar meson-exchange diagrams. We find that the hadronic rescattering diagrams can explain the recent experimental data in the vicinity of the threshold. In particular, the bump-like structure at the photon energy GeV is well explained by the rescattering amplitude in the intermediate state, which is the dominant contribution among other hadronic contributions. We also find that the hadronic rescattering diagrams are consistent with the observed spin-density matrix elements near the threshold region.
The meson is distinguished from other vector mesons, since it contains mainly strange quarks. Because of its dominant strange quark content, its decays to lighter mesons and coupling to the nucleon are known to be suppressed by the Okubo–Zweig–Iizuka (OZI) rule. In fact, the strange vector form factors of the nucleon, which is implicitly related to the meson via vector meson dominance, is reported to be rather small [1]. This large content of the meson makes the meson exchange picture unfavorable in describing photoproduction of the meson. Thus, the Pomeron [2,3] is believed to be the main contribution to photoproduction, since it explains the slow rise of the differential cross sections of photoproduction as the energy increases. However, while it is true in the higher energy regime, a recent measurement reported by the LEPS collaboration [4] shows a bump-like structure around the photon energy GeV. It seems that the Pomeron alone cannot account for this bump-like structure, and that one should consider other production mechanisms of photoproduction near the threshold energy.
So far, the theoretical understanding of the production mechanism for photoproduction can be summarized as follows:
General energy dependence of the cross sections is mainly explained by Pomeron exchange that can be taken as either a scalar meson or a vector meson with charge conjugation . While the Pomeron explains the increase of the differential cross section in the forward direction, it cannot describe the behavior of near the threshold.
The exchange of neutral pseudoscalar mesons (, ) provides a certain contribution to near the threshold but it is not enough to explain the threshold behavior of [5]. Moreover, and exchanges taken alone, being their unnatural parity nature, wrongly predict the spin-density observables and, in particular, the matrix element [6] (see Appendix A for its definition).
Usual vector meson exchanges such as and are forbidden due to their negative charge conjugations (). Otherwise, the charge conjugation symmetry will be broken.
Vector meson exchanges with exotic quantum number such as are allowed, but these vector mesons, possibly hybrid exotic mesons, are not much known experimentally [7].
Understanding this present theoretical and experimental situation in photoproduction, Ozaki et al. [8] proposed a coupled-channel effect based on the -matrix formalism. They considered the and kernels [9] in the coupled-channel formalism in addition to and . It is a very plausible idea: since the threshold energy for the is quite close to that for the bump-like structure ( GeV), the resonance may influence photoproduction. Moreover, the reaction can be regarded as a subreaction for the process together with the one in Ref. [9]. In addition, a possible nucleon resonance () with large content was also taken into account. Interestingly, the coupled-channel effects were shown to be not enough to explain the bump-like structure GeV. On the other hand, the bump-like structure was described by their possible resonance and was interpreted as a destructive interference arising from the resonance [10,11].
In the present work, we want to scrutinize in detail the nontrivial hadronic contributions arising from hadronic rescattering amplitudes in addition to Pomeron and pseudoscalar meson exchanges, by reexamining the previous work of Ref. [8]. We postulate that the Pomeron amplitude near the threshold region is suppressed, while the production mechanism there is explained by hadrons. We believe that such an attempt is legitimate because we do not know much about the Pomeron at low energies, while it is well established that it plays the dominant role at high energies. Thus we determine the threshold parameter in such a way that the Pomeron exchange becomes ineffective in the threshold energy region.
For hadronic processes, we take into account seven possible rescattering amplitudes with intermdiate , , , , , , and states. However, it is quite involved to compute these rescattering equations explicitly, so we use the Landau–Cutkosky rule [12,13], which yields the imaginary part of the rescattering amplitudes by its discontinuity across the branch cut. Though their real part will surely contribute to the transition amplitude, we will show that the imaginary part already illuminates the coupled-channel effects on the production mechanism of near the threshold. The parameters such as the coupling constants and cut-off masses of the form factors will be fixed by describing the corresponding processes and by using experimental and empirical data. Yet unknown parameters are varied as compared to the present experimental data.
In the present work, we do not consider -channel resonances, since we do not have much information on them above the threshold [7]. Moreover, it is not necessary to introduce the with large hidden strangeness, which was considered in Refs. [10,11] in an attempt to describe the LEPS data.
We will show that the hadronic mechanism with coupled-channel effects are indeed essential in explaining the recent LEPS data, which is a different conclusion from Ref. [8]. Therefore, we assert in this work that the main contribution in the vicinity of the threshold comes from the hadronic coupled-channel effects, in particular, the rescattering effects, while the Pomeron becomes important only at higher energies.
The present paper is organized as follows. In Sect. 2, we explain the basic formalism. We show how to compute the rescattering effects mentioned above. In Sect. 3, we present the numerical results such as the energy dependence of the forward cross sections, the angular distributions, and the spin observables. We also discuss how the channel can explain the bump-like structure together with the tuned Pomeron exchange. We discuss in detail the spin-density matrix elements for photoproduction. The final section is devoted to summary and outlook. In Appendix A, we present the definition of the spin-density matrix elements for reference.
General formalism
In the present work, we will employ the effective Lagrangian approach in addition to Pomeron exchange. In Fig. 1, we draw the relevant Feynman diagrams involved in describing photoproduction. The first diagram corresponds to the Pomeron exchange, and the second one depicts and exchanges. The last diagram represents generically all the contributions from various rescattering amplitudes with intermediate hadron states, i.e. , , , , , , and , among which the last one was already considered in Ref. [8]. From now on, we will simply define the rescattering amplitude as that with intermediate and states, and so on. We also define the four-momenta of the incoming photon, outgoing , the initial (target) proton, and the final (recoil) proton as and , and , respectively. In the center of mass (CM) frame, these variables are written as , , , and , where , , , and , respectively.
Relevant Feynman diagrams for photoproduction: We draw, from the left, the diffractive Pomeron exchange, the pseudoscalar meson exchanges, and the generic rescattering diagram that includes intermediate meson and baryon states.
The invariant amplitude related to the diagrams in Fig. 1 can be expressed as the Blankenbecler–Sugar (BbS) equation [14]
which was derived from the three-dimensional reduction of the Bethe–Salpeter equation. The amplitude includes all the diagrams at the tree level. On the other hand, and are the off-mass-shell extended diagrams for the and , respectively. and designate the off-mass-shell energies in the intermediate meson and baryon states as shown in the third diagram of Fig. 1: and , with the off-mass-shell three-momentum . denotes a Mandelstam variable, i.e. the square of the total energy, . The second term of Eq. (1) corresponds to the third diagram of Fig. 1. In fact, it is quite involved to solve Eq. (1) in a full coupled-channel formalism. Instead, we will concentrate on the imaginary part of Eq. (1), which can be obtained by using the two-body unitarity relation. It is much simpler than solving Eq. (1) directly, since we only need the on-shell amplitudes and . Of course, the price we have to pay is to ignore the real contributions, which may be equally as important as the imaginary ones. However, as will be shown soon, this approximation already conveys significant information on how these coupled-channel effects are essential.
Pomeron exchange
The amplitude of the Pomeron exchange [15–17] corresponding to the first diagram of Fig. 1 is given by
where and are the polarization vectors of the meson and photon, respectively. is
where the transition operator is defined as
with . Note that the Pomeron amplitude preserves gauge invariance . The corresponding invariant amplitude in Eq. (3) is written as
where and . is the isoscalar form factor of the nucleon, whereas is the form factor for the photon– meson–Pomeron vertex. They are parameterized, respectively, as
The Pomeron trajectory in Eq. (5) is determined from hadron elastic scattering in the high-energy regime, and GeV. The prefactor in Eq. (5) governs the overall strength of the amplitude and is given by , as in Ref. [8]. Since the Pomeron exchange is in fact the main contribution in the high-energy regime, while the production mechanism at low energies should be explained by hadronic degrees of freedom, we introduce a phenomenological factor to suppress the Pomeron effects. It is plausible, because it still keeps all the important high-energy behavior of the Pomeron exchage. The factor is given as
where is the threshold energy of photoproduction.
<inline-formula><tex-math notation="LaTeX"><![CDATA[$\pi $]]></tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX"><![CDATA[$\eta $]]></tex-math></inline-formula> exchanges
To calculate pseudoscalar meson exchange in the channel (the second diagram of Fig. 1), we introduce the following effective Lagrangians:
where , , and denote the vector meson, photon, and nucleon fields, respectively. and stand for the meson and nucleon masses respectively. represents the electric charge. The -channel amplitude then takes the following form:
where is the four-momentum of an exchanged pseudoscalar meson. We introduce the monopole-type form factors for each vertex and , defined as
For the coupling constants and the cut-off masses for the pseudoscalar exchange, we follow Ref. [5]: , for the and coupling constants, respectively. The cut-off masses are taken to be and . Though these values are somewhat different from the phenomenological nucleon–nucleon potentials [18,19], the effects of the pseudoscalar meson exchanges on photoproduction are rather small. Thus, we will take the values given above typically used in photoproduction. Those of the coupling constants for the vertices are determined by using the radiative decays of the meson to and . Using the data from the Particle Data Group (PDG) [7], one can find and . The negative signs of these coupling constants were determined by the phase conventions in SU(3) symmetry as well as by photoproduction [5]. We choose the cut-off masses for the and form factors as follows: GeV and GeV, respectively.
In addition to the Pomeron and pseudoscalar meson exchanges, we need to consider the seven different rescattering amplitudes. However, the corresponding transition processes are not well known phenomenologically. Thus, we construct the seven different rescattering amplitudes with the Born approximation for simplicity: , , , , , , and , based on the effective Lagrangians. Since the rescattering amplitude is the most significant among several possiblities in describing photoproduction, we first discuss the one and then deal with all the other rescattering ones in the next subsection. The process was investigated within an effective Lagrangian method in Ref. [9], with results in good agreement with the experimental data. Thus, we will employ the formalism developed in Ref. [9] so that we may take into account the coupled-channel effects more realistically.
The effective Lagrangians for are written as
where and denote the meson and fields, respectively. For , we utilize the Rarita–Schwinger formalism. is the kaon mass. The coupling constant is taken from Ref. [9], since we use the amplitude derived there. The coupling constant can be determined from the experimental data for the decay width . On the other hand, is not well known experimentally. Recent experiments measuring the strange vector form factors imply that the strange quark gives almost no contribution to the nucleon electromagnetic (EM) form factors [1]. One can deduce from this experimental fact that the coupling constant should be very small. In Ref. [21], the was estimated by using a microscopic hadronic model with continuum: and , which are compatible with the recent data for the strange vector form factors. Thus, we will take these values in the present work. However, note that the -channel contribution with the vertex is almost negligible. In Table 1, the relevant strong coupling constants and anomalous magnetic moments are listed.
The strong coupling constants and anomalous magnetic moments used in the present work.
11
Ref. [9]
4.7
Ref. [7]
0.25
Ref. [21]
1.79
Ref. [7]
0.2
Ref. [21]
Based on the effective Lagrangians given in Eq. (11), we can write down the amplitude for the rescattering equation, Eq. (1). As mentioned previously, it is rather complicated to solve the BbS equation (1). Thus, instead of dealing with Eq. (1) directly, we will use the two-body unitarity relation to further select the imaginary part of Eq. (1). This is in fact identical to using the Landau–Cutkosky rule [12,13]. Knowing the singularities and the cuts of the invariant amplitudes, and utilizing the two-body unitarity, we can easily obtain the imaginary part of the invariant amplitudes. Having computed the Lorentz-invariant phase space volume factors, we obtain the imaginary part of the amplitude as
where is the magnitude of the on-mass-shell three-momentum. This imaginary part of the amplitude is schematically drawn in Fig. 2. The shaded ellipse on the left-hand side represents the invariant amplitude for , which is basically the same as that of Ref. [9] except for different form factors, as will be explained later. It consists of three different types of the Feynman diagrams as shown below the left dashed arrow. On the other hand, the right ellipse stands for the process that contains the diagrams below the right arrow, generically. Note that we use a similar method as in Ref. [8] but we choose different form factors and parameters. The corresponding invariant amplitudes and with the form factors are defined as follows:
where (), (), and () represent the -channel, the -channel, and the contact-term contributions to the () process, respectively:
We introduce the form factors and for and , respectively, in particular, in a gauge-invariant manner for the rescattering:
where the cut-off masses and powers are fitted to the experimental data for and , which are listed in Table 2. In Fig. 3, we draw the numerical result of the total cross section for in comparison with the experimental data taken from Ref. [23]. It is in good agreement with the data.
Cut-off parameters used in Eq. (13)
6
2
1
1
1.59 GeV
0.65 GeV
1.5 GeV
1.5 GeV
Feynman diagrams for the rescattering. The form factors are introduced in a gauge-invariant way.
Total cross section of the reaction as compared to the experimental data [23].
All other rescattering amplitudes
In the same manner as done for the rescattering amplitude, we consider the six different amplitudes, i.e. , , , , , and . photoproduction has been studied theoretically in Refs. [11,24–26], in which the contributions of the -channel and exchanges were considered and exchange was found to be the dominant one, since it selects the isovector part of the EM current. Thus, we compute and with the and exchanges in the -channel, respectively. We will show later, in Fig. 5, that indeed the exchange describes well, qualitatively, the reaction. In Ref. [24], photoproduction was also discussed within the same framework. In contrast to the reaction, the exchange appeared to be dominant, since it picks up the isoscalar part of the EM current. Correspondingly, we consider the rescattering contribution, where is produced by the one pion exchange. The and rescattering amplitudes are obtained by reversing the and rescattering diagrams. The and reactions were measured by several experimental collaborations [27–32] and were investigated theoretically [33–37]. While we consider all the relevant diagrams for the rescattering contribution because of its significance, we will take into account only the -exchange diagrams in the -channel for the and rescattering diagrams, since these two rescattering diagrams turn out to have tiny effects on photoproduction.
The relevant effective Lagrangians for these rescattering diagrams are given as follows:
where the coupling constants and the cut-off masses are listed in Table 3. The invariant amplitudes for these diagrams are derived as follows:
where the subscripts correspond to the rescattering amplitudes. The other subscripts and denote the and parts, respectively. In Figs. 4 and 5 we draw the results of the total cross sections for the and reactions, respectively. The results are qualitatively in agreement with the experimental data.
Coupling constants and cut-off masses used in rescattering amplitudes
0.82
Ref. [24]
10.026
Ref. [24]
13.26
Ref. [24]
1.258
Ref. [7]
0.45
Ref. [7]
0.557
Ref. [7]
10.35
Ref. [20]
Ref. [20]
4.48
Ref. [7]
13.26
Ref. [38]
0.254 GeV
Ref. [7]
10.74
Refs. [7,39]
0
Ref. [20]
6.1
Ref. [40]
1.05 GeV
Ref. [17]
1.05 GeV
Ref. [17]
1.05 GeV
Ref. [24]
1.1 GeV
Ref. [24]
1 GeV
Ref. [24]
0.9 GeV
Ref. [24]
0.9 GeV
Ref. [7]
0.6 GeV
Ref. [20]
1.227 GeV
Ref. [41]
1 GeV
Total cross section of the reaction. The solid curve depicts the present result obtained from the -channel exchange diagram. The closed circles and the open squares are taken from Ref. [42], whereas the open triangles are taken from Ref. [43].
Total cross section of the reaction. The solid curve depicts the present result obtained from the -channel exchange diagram. The closed squares denote the experimental data from Ref. [46], whereas the open circles represent those from Ref. [44].
Numerical results and discussion
We are now in a position to discuss the numerical results for photoproduction. We start with the differential cross section at the forward angle as a function of the photon energy in the laboratory frame. The parameters are determined in the following manner. We fix the cut-off parameters for the box diagrams to describe the dependence of in the lower energy region, in particular, to explain the well-known bump-like structure around GeV. The parameters of all other hadronic diagrams are taken from existing references as explained in the previous section.
Figure 6 illustrates various contributions to as a function of the photon energy from the Pomeron exchange, the -channel and exchanges, and seven rescattering diagrams. The solid curve with symbol draws the contribution of the Pomeron exchange to . As expected, it governs dependence in the higher energy region ( GeV). Note, however, that the Pomeron does not contribute to below around in the present work. The and exchanges provide a certain amount of effect on the differential cross section (solid curve with symbol ). The contribution of the and exchanges starts to increase from the threshold energy and then it decreases very slowly when it reaches approximately 3 GeV. Thus, the effects of the and exchanges are quite important in the lower energy region up to GeV, where the Pomeron exchange overtakes the and exchanges.
Differential cross section as a function of the photon energy on a log scale. The thick solid curve depicts the result with all contributions included. The solid curves with the symbols , , , and represent the Pomeron contribution, those of and exchanges, those of all the box diagrams, and the total contribution of hadronic diagrams (), respectively. The dashed curves with numbers in order denote the effects of the seven rescattering diagrams separately.
Except for the rescattering diagram, all other rescattering contributions turn out to be negligibly small. However, the rescattering diagram plays an essential role in describing the experimental data for in the lower region, in particular, explaining the bump-like structure near GeV. In Fig. 7, we enlarge the results of Fig. 6 in the threshold region. This is very different from the conclusion of Ref. [8], where the seems to be suppressed in the -matrix formalism. The reason lies in the fact that we have considered the imaginary contribution of the coupled-channel effects through the unitarized rescattering equation (1) and have introduced different form factors for the and reactions. In general, form factors are given as functions of two Mandelstam variables for the rescattering diagrams, i.e. , since we have two off-shell particles in the -channel and another two off-shell particles in the -channel. However, it is very difficult to preserve the gauge invariance in the presence of the form factors. Thus, we have introduced a type of overall form factor to keep the gauge invariance in the part, as written in Eq. (13). To maintain consistency, we have also included a similar type of form factor in the part. With these form factors considered, we find that the rescattering diagram is indeed enhanced as shown in Fig. 6 in comparison with Ref. [8]. The contribution of the rescattering diagram increases sharply up to GeV and then falls off linearly. The result of the rescattering diagram indicates that the off-shell effects, which arise from the form factors and the rescattering equation, may come into play. Moreover, the form factors given in Eq. (13) are crucial in explaining the differential cross section as a function of which will be shown in Fig. 8. Another reason for the discrepancy may be due to fact that there is some inconsistency in Ref. [8]: the -matrix formalism that was employed in Ref. [8] is only valid for real , by definition. On the other hand, the Pomeron exchange provides only the imaginary amplitude. That is the reason why the sign had to be modified as mentioned in the erratum given in Ref. [8]. Moreover, the present way of treating the coupled-channel effects is rather different from Ref. [8].
The differential cross section as a function of the photon energy near the threshold on a linear scale.
The differential cross section as a function of the scattering angle with the photon energy at GeV. The thick solid curve depicts the result with all hadronic contributions included. The solid curves with the symbols and represent the contribution of the and exchanges and those of all the box diagrams, respectively. The dashed curves with numbers in order denote the effects of the seven rescattering channels separately.
Considering the fact that the threshold energy ( GeV) is very close to that of photoproduction ( GeV), one might ask why the contribution of the is suppressed. While the channel ( GeV) is directly related to , since both are subreactions of the process, the reaction is distinguished from those two reactions, because the channel is related to the reaction. Thus, one can qualitatively understand why the contribution of the rescattering diagrams is suppressed.
In Fig. 8, the differential cross section as a function of the scattering angle is depicted at GeV. Since the Pomeron exchange is suppressed at this photon energy because of GeV, we can examine each hadronic contribution to the differential cross section in more detail. Figure 8 clearly shows that the rescattering diagram is the most dominant among the hadronic contributions. Adding all the effects of the rescattering diagrams, we find that the rescattering contributions almost describe the dependence. Together with the and exchanges, the result of the differential cross section is in good agreement with the experimental data [4,45].
The differential cross section as a function the scattering angle is drawn in Fig. 9. The left and right panels correspond to the photon energies GeV and GeV, respectively. As expected, the hadronic contribution is dominant over the Pomeron exchange at the lower photon energy, while at GeV the Pomeron governs the process. Interestingly, the effects of the rescattering diagrams, in particular the one, turn out to be larger than those of the and exchanges, whereas the rescattering diagrams seem to be suppressed at higher photon energies. This implies that the rescattering diagram influences photoproduction only in the vicinity of the threshold energy. Figure 10 depicts the results of the differential cross section as a function of with eight different photon energies, where is the minimum four-momentum transfer from the incident photon to the meson. The results are in good agreement with the experimental data taken from the measurement of the LEPS collaboration [4].
The differential cross section as a function of the scattering angle with two different photon energies GeV and GeV. The thick solid curve depicts the result with all contributions included. The solid curves with the symbols , , , and represent the Pomeron contribution, those of the and exchanges, those of all the rescattering diagrams, and the total contribution of hadronic diagrams (), respectively.
Differential cross sections of the reaction as a function of with eight different photon energies. The experimental data are taken from Ref. [4].
It is of great importance to examine the angular distribution of the decay in the rest frame or in the Gottfried–Jackson (GJ) frame, since it makes the helicity amplitudes accessible to experimental investigation [47,48]. The detailed formalism for the angular distribution of the meson decay can be found in Refs. [5,48]. The decay angular distribution of photoproduction was measured at SAPHIR/ELSA [49], but the range of the photon energy is too wide. On the other hand, the LEPS collaboration measured the decay angular distribution at forward angles () in two different energy regions: GeV and GeV [4], which are related to the energy around the local maximum of the cross section and that above the local maximum, respectively. Therefore, we have computed the decay angular distributions at two photon energies, i.e. GeV and GeV, which correspond to the center values of the given ranges of in the LEPS experiment.
The one-dimensional decay angular distributions , , are presented in Fig. 11, and are expressed respectively as
where and denote the polar and azimuthal angles of the decay particle in the GJ frame. represents the azimuthal angle of the photon polarization in the center-of-mass frame. stands for the degree of polarization of the photon beam. , , and are defined as
The expressions for the spin-density matrix elements with the helicities and for the meson can be found in Appendix A.
The decay angular distributions for in the Gottfried–Jackson frame. We take the center values of the energy ranges measured by the LEPS collaboration [4], i.e. GeV and GeV. The experimental data are taken from Ref. [4].
Panel (a) of Fig. 11 draws the one-dimensional decay polar-angle distributions . As pointed out by Refs. [4,6], the decay distribution behaves approximately as , which indicates that the helicity-conserving processes are dominant as shown in Eq. (18). This means that -exchange particles with unnatural parity at the tree level do not contribute to . As will be discussed later, from the and exchanges, which is related to the single spin-flip amplitude in the GJ frame, exactly vanishes. On the other hand, all hadronic rescattering diagrams contribute to it. Though the Pomeron exchange might contribute to this spin-density matrix element, it does not play any role below 2.3 GeV. Panel (b) of Fig. 11 shows the results of , which are in good agreement with the LEPS data, whereas Panel (c) depicts those of , , and , respectively, which deviate from the data. In fact, the data show somewhat irregular behavior which does not seem to be easily reproduced.
As shown in Fig. 11, the decay angular distributions shed light on the production mechanism of the meson, since they make it possible to get access experimentally to the spin-density matrix elements, or the helicity amplitudes of photoproduction. It has important physical implications, because even though some diagrams seem to contribute negligibly to the cross sections, they might have definite effects on the decay angular distributions. In Table 4, the contributions of each rescattering diagram to the various spin-density matrix elements at are listed. As expected, the and exchanges contribute only to . The hadronic rescattering diagrams mainly contribute to and , and are almost negligible to other components. Interestingly, the rescattering diagram is the dominant one for , even though it provides much smaller effects on the differential cross section than the one.
density matrix in the forward scattering at GeV
-channel
0
0.5
0
0
0
0.651
0.175
0.035
0.48
0.254
0.066
0.061
0.448
0.025
0.488
0.030
0.485
0.499
rescattering
0.455
hadrons
0.24
Recently, the LEPS experiment measured the spin-density matrix elements for [6] in the range of – GeV in which the Pomeron exchange does not play a particularly important role in the present approach. Thus, we can examine the hadronic contributions to each spin-density matrix element. Figure 12 illustrates the various spin-density matrix elements, compared with the LEPS data. Since the experimental data are given in the finite range of , we just take the three center values corresponding to the ranges, i.e. , 2.07, 2.27 GeV. The hadronic diagrams considered in the present work describe quantitatively , , and . However, the deviations are found in other spin-density matrix elements as increases.
The density matrix elements as a function of for three different photon energies, i.e. GeV, GeV, and GeV, to which the solid, dotted, and dot-solid curves correspond. The experimental data with three different ranges of the photon energy are taken from Ref. [6].
Summary and outlook
In the present work, we aimed at investigating the coupled-channel effects arising from the hadronic intermediate rescattering diagrams to photoproduction near the threshold region in addition to the conventional approach of Pomeron, , and exchanges. In particular, the bump-like structure near GeV, which was reported by the LEPS experiment [4], sheds light on the production mechanism of the meson in the vicinity of the threshold, since the Pomeron exchange could not explain this peculiar structure of photoproduction. Thus, we studied in detail the effects of the seven different rescattering diagrams such as , , , , , , and with the Pomeron exchange mechanism suppressed near the threshold region. In order to take into account the rescattering terms, we employed the Landau–Cutkosky rule in dealing with these rescattering diagrams.
Since it turned out that the rescattering diagram played a dominant role among hadronic contributions in the lower-energy region, we scrutinized its contribution to photoproduction. We introduced form factors depending on both the and Mandelstam variables in such a way that the total cross section of the reaction was reproduced well. All other rescattering diagrams were constructed by utilizing the previous theoretical works and by reproducing the corresponding experimental data when they were available. We examined each contribution carefully by computing the differential cross section of photoproduction. While the rescattering diagram was found to be the most dominant near 2 GeV, all other rescattering diagrams turned out to be very small. The results were in good agreement with the LEPS data, including the bump-like structure. We also computed the differential cross section as a function of and found it to be in good agreement with the experimental data.
We investigated the contributions of hadronic rescattering diagrams to the decay angular distributions. While the one-dimensional angular distributions and were in good agreement with the experimental data, the other three angular distributions seemed to deviate from the LEPS experimental data. We also examined the various spin-density matrix elements, which were measured recently by the LEPS collaboration. We found that the hadronic rescattering diagrams describe the experimental data for , , and well. While the present results explain near relatively well for other spin-density matrix elements, they deviated from the expeimental data as increased. Therefore, the conclusion of the present work is that the production mechanism of photoproduction near the threshold is governed by the hadronic coupled-channel effects, whereas the Pomeron only becomes important in the higher energy region.
As shown in the present work, the intermediate rescattering amplitudes, in particular the one, play crucial roles in explaining the cross sections of the reaction in the vicinity of the threshold. Other rescattering diagrams also provided certain effects on the part of the spin-density matrix elements. We have considered in this work only the imaginary part of the transition amplitudes of the rescattering diagrams based on the Landau–Cutkosky rule. However, the results of the spin-density matrix elements already indicate that we should carry out a theoretical analysis of photoproduction more systematically and quantitatively. Thus, we need to investigate a full coupled-channel formalism and to solve rescattering equations (1) with the real parts of the rescattering diagrams fully taken into account. Another interesting and important problem is to extend our approach to the neutron target, since some of the considered amplitudes are isospin dependent. The corresponding works are under way.
Funding
Open Access funding: SCOAP^{3}.
Acknowledgements
The authors are grateful to S. Ozaki for valuable discussions. A.H. is supported in part by the Grant-in-Aid for Scientific Research on Priority Areas titled “Elucidation of New Hadrons with a Variety of Flavors” (E01:21105006). The work of H.Ch.K. was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant Number: 2012001083).
Appendix A. The spin-density matrix elements
<p>The spin-density matrix elements are expressed in terms of the helicity amplitudes [<xref ref-type="bibr" rid="PTU004C5">5</xref>,<xref ref-type="bibr" rid="PTU004C48">48</xref>]:
<disp-formula id="PTU004M1a"><label>(A1)</label><tex-math notation="LaTeX"><![CDATA[ \begin{align} \rho_{\lambda \lambda'}^{0} &= \frac{1}{N} \sum_{ \lambda_{\gamma}, \lambda_{i},\lambda_{f}} T_{\lambda_f, \lambda ; \lambda_i ,\lambda_{\gamma}} T_{\lambda_f, \lambda'; \lambda_i , \lambda_{\gamma}}^{*} , \notag\\ \rho_{\lambda \lambda'}^{1} &= \frac{1}{N} \sum_{ \lambda_{\gamma}, \lambda_{i},\lambda_{f}} T_{\lambda_f , \lambda ; \lambda_i , -\lambda_{\gamma}} T_{\lambda_f , \lambda' ; \lambda_i , \lambda_{\gamma}}^{*} , \notag\\ \rho_{\lambda \lambda'}^{2} &= \frac{i}{N} \sum_{ \lambda_{\gamma}, \lambda_{i},\lambda_{f}} \lambda_\gamma T_{\lambda_f , \lambda ; \lambda_i , -\lambda_{\gamma}} T_{\lambda_f , \lambda' ; \lambda_i , \lambda_{\gamma}}^{*} , \notag\\ \rho_{\lambda \lambda'}^{3} &= \frac{1}{N} \sum_{ \lambda_{\gamma}, \lambda_{i},\lambda_{f}} \lambda_\gamma T_{\lambda_f , \lambda ; \lambda_i , \lambda_{\gamma}} T_{\lambda_f , \lambda' ; \lambda_i , \lambda_{\gamma}}^{*} , \label{eqA1} \end{align} ]]></tex-math>
</disp-formula>
where <inline-formula><tex-math notation="LaTeX"><![CDATA[$\lambda _\gamma $]]></tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX"><![CDATA[$\lambda _i$]]></tex-math></inline-formula>, and <inline-formula><tex-math notation="LaTeX"><![CDATA[$\lambda _f$]]></tex-math></inline-formula> represent the helicities for the photon and the initial and final nucleons, respectively, whereas <inline-formula><tex-math notation="LaTeX"><![CDATA[$\lambda $]]></tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX"><![CDATA[$\lambda '$]]></tex-math></inline-formula> denote those for the <inline-formula><tex-math notation="LaTeX"><![CDATA[$\phi $]]></tex-math></inline-formula> meson. The normalization factor <inline-formula><tex-math notation="LaTeX"><![CDATA[$N$]]></tex-math></inline-formula> is defined as
<disp-formula id="PTU004M2a"><label>(A2)</label><tex-math notation="LaTeX"><![CDATA[ \begin{equation}\label{eqA2} N = \sum|T_{\lambda_f,\lambda;\lambda_i, \lambda_{\gamma}}|^2. \end{equation} ]]></tex-math>
</disp-formula>
</p>
</sec>
</app>
</app-group>
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